## HOW TO CONVERT PURE SURD INTO MIXED SURD

On the webpage "how to convert pure surd into mixed surd" we are going to see the steps involved to convert pure surd into mixed surd.

## What is surd?

A number which cannot be simplified to remove the square root or cube root.

 Examples Simplified value Surd or not √2√4∛27 √2√4 = √2 x 2 = 2∛3 x 3 x 3 = 3 SurdNot surdNot Surd

In the above example √2 is surd, because we cannot simplify √2 hereafter.But √4 and ∛27 are not surds, because we can simplify those numbers hereafter.

## Definition of Pure surds

A surd in which the whole of the rational number is under the radical sign and makes the radicand, is called pure surd.

In other words a surd having no rational factor except unity is called a pure surd or complete surd.

For example, each of the surds √7, √10,

## Definition of mixed surds

A surd having a rational co-efficient other than unity is called a mixed surd.

In other words if some part of the quantity under the radical sign is taken out of it, then it makes the mixed surd.

For example, each of the surds 2√7

In the following example we are going to learn how to simplify radical expressions.

Problem 1:

Simplify the following radical expression

√27 + 5 √75 + √108 -3 √48

Solution:

= √27 + 5 √75 + √108 - 3 √48

First we have to split the given numbers inside the radical as much as possible.

=  3 √3 + 25 √3 + 6 √3 - 4 √3

= (3 + 25 + 6 - 4) √3

= 30 √3

Now let us see the next example of the topic "how to convert pure surd into mixed surd".

Problem 2:

Simplify the following radical expression

7 √30 + 2 √75 + 5 √50

Solution:

= 7 √30 + 2 √75 + 5 √50

First we have to split the given numbers inside the radical as much as possible.

=  √(5 x 2 x 3) + √(5 x 5 x 3) + √(5 x 5 x 2)

Here we have to keep √30 as it is.

=  √30 + 5 √3 + 5 √2

Now let us see the next example of the topic "how to convert pure surd into mixed surd".

Problem 3:

Simplify the following radical expression

√27 + √105 + √108 + √45

Solution:

= 3 √5 + 2√95 + 3√117 - √78

First we have to split the given numbers inside the radical as much as possible

=  √(3 x 3 x 3) + √(5 x 3 x 7) +

√(3 x 3 x 3 x 2 x 2) - √(5 x 5 x 3)

=  3 √3 +  √105 + 3 x 2 √3 - 5 √3

=  3 √3 +  √105 + 6 √3 - 5 √3

= (3 + 6 - 5) √3 + √105

= 4 √3 + √105

Now let us see the next example of the topic "how to convert pure surd into mixed surd".

Problem 4:

Simplify the following radical expression

√45 + 3 √20 + √80 - 4 √40

Solution:

= √45 + 3 √20 + √80 - 4 √40

First we have to split the given numbers inside the radical as much as possible.

=  √(3 x 3 x 5) + √(2 x 2 x 5) +

√(5 x 2 x 2 x 2 x 2) - √(5 x 2 x 2 x 2)

=  3 √5 + 2 √5 + 2 x 2 √5 - 2 √(2 x 5)

=  3 √5 + 2 √5 + 4 √5 - 2 √10

= (3 + 2 + 4) √5 - 2 √10

= 9 √5 - 2 √10

Now let us see the next example of the topic "how to convert pure surd into mixed surd".

Problem 5:

Simplify the following radical expression

3√5 + 2√95 + 3√117 - √78

Solution:

= 3 √5 + 2√95 + 3√117 - √78

First we have to split the given numbers inside the radical as much as possible

=  3 √5 + 2 √(5 x 19) + 3 √(3 x 3 x 13) - √(3 x 2 x 13)

=  3 √5 + 2 √95 + 3 x 3 √13 - √78

=  3 √5 + 2 √95 + 9 √13 - √78

Now let us see the next example of the topic "how to convert pure surd into mixed surd".

Problem 6:

Simplify the following radical expression

3 √32 - 2√8 + √50

Solution:

= 3 √32 - 2 √8 + √50

First we have to split the given numbers inside the radical as much as possible.

=  3 √(2 x 2 x 2 x 2 x 2) - 2 √(2 x 2 x 2) + √(5 x 5 x 2)

=  (3 x 2 x 2 )√2 - (2 x 2) √2 + 5 √2

=  12 √2 - 4 √2 + 5 √2

= (12 + 5 - 4) √2

= 13 √2

Now let us see the next example of the topic "how to convert pure surd into mixed surd".

Problem 7:

Simplify the following radical expression

2 √12 - 3√27 - √243

Solution:

= 2 √12 - 3 √27 - √243

First we have to split the given numbers inside the radical as much as possible.

= 2 √(2 x 2 x 3) - 3 √(3 x 3 x 3) - √(3 x 3 x 3 x 3 x 3)

=  (2 x 2) √3 - (3 x 3) √3 - (3 x 3) √3

=  4 √3 - 9 √3 - 9 √3

= ( 4 - 9 - 9 ) √3

= -14 √3

Now let us see the next example of the topic "how to simplify radical expressions".

Problem 7:

Simplify the following radical expression

√54 - √2500 - √24

Solution:

= √54 - √2500 - √24

First we have to split the given numbers inside the radical as much as possible.

=  √(2 x 3 x 3 x 3) - √(5 x 5 x 5 x 5 x 2 x 2) -

√(3 x 2 x 2 x 2)

=  3 √(3 x 2) - (5 x 5 x 2) - (2 x 2) √(2 x 3)

=  3 √6 - 50 - 4 √6

=  (3 - 4) √6 - 50

=  -√6 - 50

Now let us see the next example of the topic "how to simplify radical expressions".

Question 9

Simplify the following radical expression

√45 - √25 - √80

Solution

=  √(5 x 3 x 3) - √(5 x 5) - √(5 x 2 x 2 x 2 x 2)

=  3 √5 - 5 - 2 x 2√5

=  3 √5 - 5 - 4√5

=  -√5 - 5

Now let us see the next example of the topic "how to convert pure surd into mixed surd".

Problem 10:

Simplify the following radical expression

5√95 - 2√50 - 3√180

Solution

= 5 √95 - 2 √50 - 3 √180

First we have to split the given numbers inside the radical as much as possible.

=  5 √95  -  2 √(2 x 5 x 5) - 3 √(3 x 3 x 2 x 2 x 5)

=  5 √95 - (2 x 5) √2 - (3 x 2 x 3 )√5

=  5 √95 - 10 √2 - 18 √5

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